Question: All of the 5th grade teachers and students from Oak went on a field trip to an art museum. Tickets were $$7.00$ each for teachers and $$4.50$ each for students, and the group paid $$64.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$28.00$ each for teachers and $$10.50$ each for students, and the group paid $$196.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7x+4.5y = 64}$ ${28x+10.5y = 196}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-28x-18y = -256}$ ${28x+10.5y = 196}$ Add the top and bottom equations together. $ -7.5y = -60 $ $ y = \dfrac{-60}{-7.5}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {7x+4.5y = 64}$ to find $x$ ${7x + 4.5}{(8)}{= 64}$ $7x+36 = 64$ $7x = 28$ $x = \dfrac{28}{7}$ ${x = 4}$ You can also plug ${y = 8}$ into $ {28x+10.5y = 196}$ and get the same answer for $x$ ${28x + 10.5}{(8)}{= 196}$ ${x = 4}$ There were $4$ teachers and $8$ students on the field trips.